$2.(20)$ There are two metal electrodes with nanoscale air gap of $\backslash ($ L $\backslash ).$ The work function of two same metals are W$.$

$(1)$ Draw the energy diagram for electrons of kinetic energy E inside the metals, as shown in figure. Represent the work function W in the figure.

$(2)$ If $\backslash (\backslash$mathrm$\{$W$\}=6\backslash$mathrm$\{$eV$\}\backslash )$ and the air gap is $\backslash (\backslash$mathrm$\{$L$\}=1\backslash$mathrm$\{$~nm$\}\backslash ),$ calculate the tunneling probability of electrons between the two metals.

$(3)$ To make the tunneling probability as $\backslash (10^\{-5\}\backslash )$ in $(2),$ find how much voltage we need to apply between two metals.